What is the Correct Form of the Free Propagator in QFT?

[Phymath]

Homework Statement

from Zee QFT in a nutshell

the free propagator between two "sources" on the field is given by$$D(x_\mu) = -i \int \frac{d^3k}{(2\pi)^3 2 \omega_k}[e^{-i(\omega_kt-k\bullet x)} \Theta(x_0) + e^{i(\omega_k t-k\bullet x)} \Theta(-x_0)$$

for a space like separation ($$x_0 = 0$$) Zee gets

$$-i\int\frac{d^3k}{(2\pi)^3 2 \omega_k}e^{-i k\bullet x}$$

with assumption that $$\Theta(0) = 1/2$$

with that assumption i don't agree with Zee i get

$$-i\int\frac{d^3k}{(2\pi)^3 2 \omega_k}cos(k \bullet x)$$

where am I going wrong?

 

[Avodyne]

The two expressions are equal. If you write the complex exponential as a sum of a sine and cosine, the sine term will integrate to zero because it is odd in k.

 

[mpampinos]

In the same book, in this definition of the D(x). Why do we get a term exp^-i(ωt-kx) when X_o in positive and a term exp^i(ωt-kx) when X_o is negative?